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Sequences and Series

Short Answer Type

131. Find the sum to n terms of the following series.

space space space 1 cross times 2 cross times 3 plus 2 cross times 3 cross times 4 plus 3 cross times 4 cross times 5 plus........

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132. Find the sum of n terms of the series 3 + 7 + 13 + 21 + 31 +.......

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133. The ages of the students in a class form an A.P. whose common difference is 4 months. If the youngest student is 8 years old and the sum of the ages of all the students in the class is 168 years. Find the numbers of students in the class.

Let the number of students in the class be n.

Let a be the age of youngest student

∴ a = 8 years

Since the age of the students form an A.P. with common difference 4 months or $1 third$ years.

∴ $straight d equals space 1 third$ years

Since the sum of the ages of all the students in the class is 168 years.

∴ $straight S subscript straight n space equals space 168$ $rightwards double arrow space$ $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$

$rightwards double arrow$ $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$ $rightwards double arrow$ $16 straight n plus fraction numerator straight n squared minus straight n over denominator 3 end fraction space equals space 336$

$rightwards double arrow$ $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$ $rightwards double arrow$ $straight n squared plus 47 straight n minus 1008 space equals space 0$

$rightwards double arrow$ $space space straight n space equals space fraction numerator negative 47 plus-or-minus square root of left parenthesis 47 right parenthesis squared plus 4 left parenthesis 1 right parenthesis left parenthesis 1008 right parenthesis end root over denominator 2 left parenthesis 1 right parenthesis end fraction space equals space fraction numerator negative 47 plus-or-minus square root of 2209 plus 4032 end root over denominator 2 end fraction$

 $space space equals space fraction numerator negative 47 plus-or-minus square root of 6241 over denominator 2 end fraction space equals space fraction numerator negative 47 plus-or-minus 79 over denominator 2 end fraction space equals space 32 over 2 comma space minus 126 over 2 space equals space 16 comma space minus 63$

$rightwards double arrow$ n = 16

∵ Number of students cannot be negative. Hence, number of students in the class is 16.

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134.

A man deposited Rs. 10,000 in a bank at the rate of 5% simple interest annually. Find the amount in 15th year since he deposited the amount and also calculate the total amount after 20 years.

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135. An insect starts from a point and travels in a straight path 1 mm in the first second and half of the distance covered in the previous second in the succeeding second. In what time would it reach a point 3 mm away from its starting point?

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136. The interior angles of a polygon are in A.P. The smallest angle is 120° and the common difference is 5°. Find the number of sides of the polygon.

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Long Answer Type

137. Shamshad Ali buys a scooter for Rs 22000. He pays Rs. 4000 cash and agrees to pay the balance in annual instalment of Rs 1000 plus 10% interest on the unpaid amount. How much will the scooter cost him?

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Short Answer Type

138. A man buys every year Bank’s cash certificates of value exceeding the last year purchase by Rs. 25. After 20 years, he finds that the total value of the certificates purchased by him is Rs. 7250. Find the values of the certificates purchased by him

(i) In the first year. (ii) In the 13th year.

179 Views

Long Answer Type

139. An odd number of stones lie along a straight path, the distance between any two consecutive stones being 10 m. The stones are to the collected at the place where the middle stone lies. A man can carry only one stone at a time. He starts carrying the stones beginning from the extreme stone. If he covers a path of 3 km, how many stones are there ?

183 Views

Short Answer Type

140. The income of a person is Rs. 3,00,000 in first year and he receives an increase of Rs. 10,000 to his income per year for next 19 years. Find the total amount, he received in 20 years.

126 Views